Sharing animation clips between similar rigs

I'm making a game in which there are several characters with similar rigs, who should be able to use the same animations. The characters, rigs and animations are made in Blender and imported into Unity as .FBX files.

Everything works fine for the first character, imported with its animations. But importing every character with all animations from Blender seems a waste of time and data, since those animations are the same : so I decided to import only Character A with all animations, and all the rest (Character B, C etc.) with only the rig, and share Character A's animations.

But : even when Character B is equipped with the same Animator Controller as Character A, nothing happens : no animation clip plays at all, despite the fact the the States in the Animator happen correctly (e.g. the Character B object moves on scene, the Animator Controller displays the progress bar moving in the "walking" state, but Character B objet shows no animation at all).

As far as I researched, it should work as long as all characters have the same rig with identical bone names, which is the case, so I don't understand what's wrong. Could the problem be that the Character A is imported along with a child object which has its own rig (a coffee cup with one single bone) ? If that's not the problem I'd like to keep my coffee cup...